Understanding the Effect of the Beta Parameter on Demonstration Testing

Reliability demonstration tests are some of the most commonly performed tests in reliability programs. Typically set up as zero failure tests, their
design involves determining the required sample size and test time per unit in order to demonstrate a component's target reliability at a desired confidence level.
In many cases, one of the inputs required to design such a test is the underlying failure distribution of the component along with a parameter of that distribution.
If the Weibull distribution is used, this parameter is the shape parameter beta. Some questions one might ask are "What is the effect of the assumed beta on
the test design?" and "What values will produce more conservative results?" This article will attempt to answer these questions through
a test design example.

Reliability Demonstration Tests

As discussed in the December 2010 issue of HotWire,
one of the most common methods for demonstration test design is based on the cumulative binomial distribution:

(1)

where:

CL is the confidence level.

Rdemo is the reliability to be demonstrated.

n is the required sample size.

r is the number of failures allowed during the test.

In the case where the available test time is the same as the time at which the target reliability should be demonstrated, Eqn. (1)
can be used to determine the required sample size. This method is referred to as a Non-Parametric Binomial test design.

If, however, you desire to either increase the sample size in order to reduce the required test time or increase the test time in
order to reduce the required sample size, then you will use the Parametric Binomial method. In this case, an underlying failure distribution
needs to be assumed, such as the Weibull distribution, along with a parameter of that distribution (the beta parameter when using the
Weibull distribution). Ideally, the value used for beta should be determined based on historical data of an identical or similar
component design. However, in many cases historical data is either not available or is of limited use and you want to understand which
assumed values of beta will produce more conservative results. We will explore this through a simple demonstration test design example.

Example

You have been required to design a demonstration test to show that a component's reliability is 90% at 1,000 hours with 90% confidence
and you have been allocated a sample size of 10 units. Historical data from similar designs have shown that the failure distribution of the
component follows the Weibull distribution with the beta parameter value ranging between 1.5 and 2.

You begin by designing a zero failure test that assumes a beta of 2. Using the RDT tool in Weibull++,
you find that the required test time per unit is 1,478 hours, as shown next.

Next, in order to investigate the effect of the assumed beta, you change its value to 1.5. In this case, you find that the required
test time per unit is 1,684 hours, as shown next.

Therefore, the conclusion is that a lower beta will result in an increased test time and produce a more conservative test plan.

You then assume that more resources become available and that the sample size increases to 30 units. If you redesign the test with
the increased sample size assuming a beta of 2, then you find that the required test time per unit is 854 hours, as shown next.

If you again change the beta parameter to 1.5, you find that the test time per unit is 810 hours, as shown next.

Therefore, in the case with a larger sample size, the effect of the beta parameter is reversed and a larger value of beta
produces the more conservative results.

The question is at what sample size does the relationship between the assumed beta parameter and the required test time reverse?
To answer this, we will use the cumulative binomial distribution equation, Eqn. (1), using a zero failure test with an underlying Weibull distribution.
In this case, Eqn. (1) becomes:

(2)

where:

CL is the desired confidence level.

ttest is the required test time.

n is the required sample size.

β is the shape parameter of the Weibull distribution.

η is the scale parameter of the Weibull distribution.

If we solve Eqn. (2) for ttest then:

(3)

Furthermore, if we solve the Weibull reliability equation for η we have:

If we substitute the above equation into Eqn. (3) and take the natural logarithm on both sides then:

From the previous equation we can see that:

If
then .

If
then .

As a result, we can conclude that:

If then
as β increases the required test
time (ttest) will also increase.

If then
as β increases the required test
time (ttest) will decrease.

Using the values from our example:

(4)

We can see that the required sample size is 22 units.

We should note that Eqn. (4) represents the calculated sample size when using the non-parametric binomial test design. As shown in the following figure,
this is confirmed where we use a non-parametric binomial test design to demonstrate 90% reliability with 90% confidence.

Therefore, we can conclude that if the available sample size is larger than the required sample size found from the non-parametric binomial
test design, then a larger beta parameter will result in a higher required test time and produce more conservative results. On the other
hand, if the available sample size is smaller than the required sample size that can be found from the non-parametric binomial test design,
then a smaller beta parameter will produce more conservative results. When the sample size is close to the required sample size found
from the non-parametric binomial test design, then the assumed beta value will only slightly affect the test time.

Conclusions

In this article we investigated the effect of the assumed beta parameter in a parametric binomial test design and found that for different
sample sizes this effect can vary. As a result, and for cases where no or limited historical data is available, one can now understand which
assumptions will produce a more conservative test plan. The example covered in this article was for the case of a zero failure test but the
same rules will also apply for cases where one or more failures are allowed during the test.